Monthly climatologies
Daily climatologies are calculated for each day-of-year (DOY), where January 1 is Day 1, January 2 is Day 2,…, December 31 is Day 366. Instead of eliminating February 29 (Day 60) on leap years, we simply skip Day 60 in years that are not leap years by adding 1 to the DOY of every day after February 28. Days start at midnight (00:00) and go until 23:59 (11:59 PM) local time.
DOY Daily Average
The Day of Year (DOY) daily average is the average of all daily averages for any given DOY. It is calculated by taking the daily averages calculated using ?@eq-tda, grouping them by DOY, and calculating the average for each DOY.
Let \(\textbf{A}\) be a matrix of daily averages from ?@eq-tda arranged with years in rows and DOY in columns, and for simplicity, let subscript \(y\) indicate year and \(d\) indicate DOY:
\[\textbf{A} = (a_{yd}) = \left( \begin{array}{cccc} a_{1,1} & a_{1,2} & ... & a_{1,d} \\ a_{2,1} & a_{2,2} & ... & a_{2,d} \\ \vdotswithin{=} & \vdotswithin{=} & & \vdotswithin{=} \\ a_{y,1} & a_{y,2} & ... & a_{y,d} \end{array} \right) \tag{1}\]
The DOY daily average is the average over each column:
\[\text{DOY daily average} = \frac{1}{Y}\sum_{y=1}^Y a_{yd} \tag{2}\]
where \(Y\) is the number of years in the observational time series.
Note: The Daily Average reported in the statistics dashboard are DOY daily averages.
Record High Daily Average
The record high daily average is the maximum daily average for any given DOY. It is determined by taking the daily averages calculated using ?@eq-tda, grouping them by DOY, and finding the maximum value for each DOY. Given \(\textbf{A}\) from Equation 1 above:
\[\text{record high daily average} = {\max}_d |a_{yd}| = \max\{|a_{yd}|: d=1,2,...,366\} \tag{3}\]
Record Low Daily Average
The record low daily average is the minimum daily average for any given DOY. It is determined by taking the daily averages calculated using ?@eq-tda, grouping them by DOY, and finding the minimum value for each DOY. Given \(\textbf{A}\) from Equation 1 above:
\[\text{record low daily average} = {\min}_d |a_{yd}| = \min\{|a_{yd}|: d=1,2,...,366\} \tag{4}\]
Average High
The average high is the average of all daily highs for any given DOY. It is determined by taking the daily highs calculated using ?@eq-dh, grouping them by DOY, and calculating the average for each DOY.
Let \(\textbf{H}\) be a matrix of daily highs from ?@eq-dh arranged with years in rows and DOY in columns, and for simplicity, let subscript \(y\) indicate year and \(d\) indicate DOY:
\[\textbf{H} = (h_{yd}) = \left( \begin{array}{cccc} h_{1,1} & h_{1,2} & ... & h_{1,d} \\ h_{2,1} & h_{2,2} & ... & h_{2,d} \\ \vdotswithin{=} & \vdotswithin{=} & & \vdotswithin{=} \\ h_{y,1} & h_{y,2} & ... & h_{y,d} \end{array} \right) \tag{5}\]
The average high is the average over each column:
\[\text{average high} = \frac{1}{Y}\sum_{y=1}^Y h_{yd} \tag{6}\]
where \(Y\) is the number of years in the observational time series.
Record High
The record high is the maximum daily high for any given DOY. It is determined by taking the daily highs calculated using ?@eq-dh, grouping them by DOY, and finding the maximum value for each DOY. Given \(\textbf{H}\) from Equation 5 above:
\[\text{record high} = {\max}_d |h_{yd}| = \max\{|h_{yd}|: d=1,2,...,366\} \tag{7}\]
Lowest High
The lowest high is the minimum daily high for any given DOY. It is determined by taking the daily highs calculated using ?@eq-dh, grouping them by DOY, and finding the minimum value for each DOY. Given \(\textbf{H}\) from Equation 5 above:
\[\text{lowest high} = {\min}_d |h_{yd}| = \min\{|h_{yd}|: d=1,2,...,366\} \tag{8}\]
Average Low
The average low is the average of all daily lows for any given DOY. It is determined by taking the daily lows calculated using ?@eq-dl, grouping them by DOY, and calculating the average for each DOY.
Let \(\textbf{L}\) be a matrix of daily lows from ?@eq-dl arranged with years in rows and DOY in columns, and for simplicity, let subscript \(y\) indicate year and \(d\) indicate DOY:
\[\textbf{L} = (l_{yd}) = \left( \begin{array}{cccc} l_{1,1} & l_{1,2} & ... & l_{1,d} \\ l_{2,1} & l_{2,2} & ... & l_{2,d} \\ \vdotswithin{=} & \vdotswithin{=} & & \vdotswithin{=} \\ l_{y,1} & l_{y,2} & ... & l_{y,d} \end{array} \right) \tag{9}\]
The average low is the average over each column:
\[\text{average low} = \frac{1}{Y}\sum_{y=1}^Y l_{yd} \tag{10}\]
where \(Y\) is the number of years in the observational time series.
Highest Low
The highest low is the maximum daily low for any given DOY. It is determined by taking the daily lows calculated using ?@eq-dl, grouping them by DOY, and finding the maximum value for each DOY. Given \(\textbf{L}\) from Equation 9 above:
\[\text{highest low} = {\max}_d |l_{yd}| = \max\{|l_{yd}|: d=1,2,...,366\} \tag{11}\]
Record Low
The record low is the minimum daily low for any given DOY. It is determined by taking the daily lows calculated using ?@eq-dl, grouping them by DOY, and finding the minimum value for each DOY. Given \(\textbf{L}\) from Equation 9 above:
\[\text{record low} = {\min}_d |l_{yd}| = \min\{|l_{yd}|: d=1,2,...,366\} \tag{12}\]